Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,13,16)(11,12,15,14)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25);(9,10)(11,14)(12,15)(13,16)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32);(2,6)(4,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32);(1,3)(2,4)(5,8)(6,7)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,16)(13,14)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72263507, #72319750, #72319776, … |
| maximal normal subgroup of | #72263507, #72319750, #72319776, … |
| maximal quotient of | #72263543, #72326825, #72326826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8362950);